Solve for $x$ : $ 8|x - 5| - 4 = -5|x - 5| + 9 $
Explanation: Add $ {5|x - 5|} $ to both sides: $ \begin{eqnarray} 8|x - 5| - 4 &=& -5|x - 5| + 9 \\ \\ { + 5|x - 5|} && { + 5|x - 5|} \\ \\ 13|x - 5| - 4 &=& 9 \end{eqnarray} $ Add ${4}$ to both sides: $ \begin{eqnarray} 13|x - 5| - 4 &=& 9 \\ \\ { + 4} &=& { + 4} \\ \\ 13|x - 5| &=& 13 \end{eqnarray} $ Divide both sides by ${13}$ $ \dfrac{13|x - 5|} {{13}} = \dfrac{13} {{13}} $ Simplify: $ |x - 5| = 1$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 5 = -1 $ or $ x - 5 = 1 $ Solve for the solution where $x - 5$ is negative: $ x - 5 = -1 $ Add ${5}$ to both sides: $ \begin{eqnarray} x - 5 &=& -1 \\ \\ {+ 5} && {+ 5} \\ \\ x &=& -1 + 5 \end{eqnarray} $ $ x = 4 $ Then calculate the solution where $x - 5$ is positive: $ x - 5 = 1 $ Add ${5}$ to both sides: $ \begin{eqnarray} x - 5 &=& 1 \\ \\ {+ 5} && {+ 5} \\ \\ x &=& 1 + 5 \end{eqnarray} $ $ x = 6 $ Thus, the correct answer is $x = 4 $ or $x = 6 $.